The answer is **Yes**. This is not really a problem about matrices. The best way to analyse it is to rewrite it in terms of the row vectors $u_i\in{\mathbb C}^n$. Let me denote $\|\cdot\|_p$ the $\ell^p$-norm over ${\mathbb C}^n$ ; when $p=2$, this is the standard Euclidian norm. The Frobenius norm is $\sum_i\|u_i\|_2^2$, while $\|A\|_\infty$ is $\max_i\|u_i\|_1$. Finally, you want to estimate $\sum_i\|u_i\|_\infty$.

Now, remember that $\|u\|_2^2\le\|u\|_1\|u\|_\infty$. Say that $\|A\|_2^2\ge\frac nc$ and $\|A\|_\infty\le c$ for some $c<\infty$. One deduces
$$\frac nc\le c\sum_i\|u_i\|_\infty.$$
By Cauchy-Schwarz, there comes
$$\frac{n^2}{c^2}\le c^2n\sum_i\|u_i\|_\infty^2,$$
which is the desired conclusion.

To summarize the calculus above, one has
$$\|A\|^2_F\le\|A\|_\infty\sum_i\|u_i\|_\infty,$$
hence
$$\|A\|_F^4\le n\|A\|_\infty^2\sum_i\|u_i\|^2_\infty.$$